Get it out of the sun. They need filtered light like the light that gets through the rain forest canopy back home.

More will be along shortly to tell you more but don't let it sunburn is what I'll say 1st! :)

Welcome to GardenWeb! Agreed, it is sunburned, these plants burn very easily.

Does the pot have a drain hole in the bottom?

"I have moisture gnats." That's a sign the soil is staying too soggy, too often. This sogginess is harmful to roots, which need air as well as moisture, or they will rot.

"For the last 2 waterings, I added dish soap (1T) and vinegar (1t)." Vinegar is often recommended as a "home remedy" for killing weeds, not something I would put anywhere near plants I like.

"A few days ago, I sprayed with an insecticide supposedly safe for indoor plants." Those are usually made to eradicate the pests that kill plants, not necessarily gnats living/breeding in the soil of the plant. It might be a fine remedy, but unless it's recommended for that purpose on the label, I wouldn't use it for that, just for what it says it's supposed to do.

"A day or two later, the flower completely drooped." Individual flowers don't last forever, so not necessarily a sign of anything except that it would be more natural for it to just turn brown, not droop.

"Now the leaves are as well." In conjunction with having gnats, I fear the roots of this plant are rotting significantly, thus unable to deliver water to the foliage even though there is plenty (too much) in the pot.

With all that's going on with this soil, I would take the plant out, rinse the old soil off of the roots, and repot with all new stuff, making sure it's in a pot with a hole in the bottom. Don't pack the soil tightly around the roots, and water when it seems like it's getting dry, not just because it's been a certain amount of time. As plants grow, and as the weather changes temp and humidity, the wind blows less or more, plants use water at different rates.

If your soil has air pockets throughout, and doesn't hold onto excess moisture, and is not constantly remoistened, fungus gnats will be unable to breed in it. This is also what is preferred by the plants, so it's a win-win thing to do.

I have a few similarly sized lily's that were experiencing the same issue. I was watering in the center of the plant which seemed reasonable. It was working well for all the other peace lily's. That is until one then two of my lily's started to droop. The fix for my problem is to water thoroughly throughout the pot while focusing on the edges. Maybe this will fix your problem too.